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Properties

Label 2-210-35.24-c2-0-2
Degree $2$
Conductor $210$
Sign $-0.981 - 0.193i$
Analytic cond. $5.72208$
Root an. cond. $2.39208$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 + 0.707i)2-s + (−0.866 + 1.5i)3-s + (0.999 − 1.73i)4-s + (2.84 − 4.11i)5-s − 2.44i·6-s + (−6.72 + 1.95i)7-s + 2.82i·8-s + (−1.5 − 2.59i)9-s + (−0.573 + 7.04i)10-s + (−8.81 + 15.2i)11-s + (1.73 + 2.99i)12-s + 7.10·13-s + (6.84 − 7.14i)14-s + (3.70 + 7.82i)15-s + (−2.00 − 3.46i)16-s + (−6.31 + 10.9i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.288 + 0.5i)3-s + (0.249 − 0.433i)4-s + (0.568 − 0.822i)5-s − 0.408i·6-s + (−0.960 + 0.279i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.0573 + 0.704i)10-s + (−0.801 + 1.38i)11-s + (0.144 + 0.249i)12-s + 0.546·13-s + (0.489 − 0.510i)14-s + (0.247 + 0.521i)15-s + (−0.125 − 0.216i)16-s + (−0.371 + 0.643i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.981 - 0.193i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.981 - 0.193i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0377797 + 0.387202i\)
\(L(\frac12)\) \(\approx\) \(0.0377797 + 0.387202i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.707i)T \)
3 \( 1 + (0.866 - 1.5i)T \)
5 \( 1 + (-2.84 + 4.11i)T \)
7 \( 1 + (6.72 - 1.95i)T \)
good11 \( 1 + (8.81 - 15.2i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 7.10T + 169T^{2} \)
17 \( 1 + (6.31 - 10.9i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (9.07 - 5.23i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (28.8 - 16.6i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 43.5T + 841T^{2} \)
31 \( 1 + (16.3 + 9.43i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-44.7 + 25.8i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 75.0iT - 1.68e3T^{2} \)
43 \( 1 - 19.6iT - 1.84e3T^{2} \)
47 \( 1 + (39.0 + 67.6i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-23.9 - 13.8i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-75.3 - 43.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (59.6 - 34.4i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-12.5 - 7.25i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 75.4T + 5.04e3T^{2} \)
73 \( 1 + (25.6 - 44.4i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (38.0 + 65.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 61.5T + 6.88e3T^{2} \)
89 \( 1 + (23.4 - 13.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 124.T + 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.78311037036871833455159601876, −11.51912479462385915874212354159, −10.15856980596314496843341872175, −9.763715637339013769260986225390, −8.843581511975351275914424698423, −7.68488930610102469619967645847, −6.26481768439057382115276531419, −5.51210255738201125735209394605, −4.13716774903826896465124213760, −2.02325918182109315829373817585, 0.25440803379963461649886122464, 2.35710818756935312411091486735, 3.51145463055122081428073658821, 5.78910267418120739627559410462, 6.53616513196759296050234427821, 7.60334893148971304004054530127, 8.782783402596807712037179754428, 9.880265788746665208922850761198, 10.80033727932153055967618435536, 11.31833615074444832026268055139

Graph of the $Z$-function along the critical line